How to Find the Radius of a Sphere Using Equations?

[cdhotfire]

Took it today, whew that was tough, anyone know where i can get some practice on the problems, i would like to do a lot better next time around. :P
Or you can post what u though of the test. Personally i though it was tough, saddly i only awnsered 9 of them . Also if I am not suposed to talk about it tell me. But i was told that i could talk about it after the end of school so, ya.

 

[Justin Lazear]

A sample test is usually included when the school receives the test booklets.

Is the cutoff for the AIME still 11 right and 0 wrong? That'd mean you're only 2 away!

--J

 

[DirtyDan]

I took it and only had 8 right (answered 10). Some other kid I know answered 17/17 right. We got to see that answer sheet afterwards, so I checked it out. Some of those problems were easy I just approached it wrong. Others were probably just as easy, accept I never grasped the concept. Did you take the AMC 12? On 16 I realized after the test I could have measured their drawing to get the answer

Oh and on 22, what does inscribed mean?

Heres that question and another if anyone else can explain

22. A rectangle P is inscribed(?) in a sphere of radius r. The surface area of P is 384, and the sum of the lengths of its 12 edges is 112. What is r?

a. 8 b. 10 c. 12 d. 14

...Would R be half of the length of the rectangle? I didnt go on because I wasnt sure what exactly "inscribed" entailed

25. Let S be the set of all points with coordinates (x,y,z), where x,y, and z are each chosen from the set {0,1,2}. How many equilateral triangles have all of their vertices in S?

a. 72 b. 76 c. 80 e. 88

I have the right answers by the way

 

[Hessam]

yep... i took it too

i believe i got 12 right out of the 14 i answered... so thus my score is a 99.5 (100 is the cut-off point for the AIME) heh... i guess i could take the 12B and try and make it again...?

I feel like this AMC was harder than the past ones

my friend answered 21 and got 20 right... this kid is a beast!

 

[cdhotfire]

I think a good thing to know for this test is dice, they love putting problems about dice, this year like 3, last year they put some also. The hardest one i did was #15, anyone get the awnser to that one, i got E. Also i hate when they ask u how many of something there are, I am like what the hell u want me to take an hour counting them, grrr. Like #11, why in the world would i spend the whole test time on that question, onless anyone knows and easy way out.

Last year my friend got 18/18, but this kid was a monster, he was so crazy while he was taking it he drank a whole Mountain Dew bottle, a 2 liter one!

 

[cdhotfire]

On 16 I realized after the test I could have measured their drawing to get the answer

What do you mean measure it, they didnt provide me with rulers, they only said use a calculator, and scrap paper!. No fair,

 

[hypermorphism]

I took it and only had 8 right (answered 10). Some other kid I know answered 17/17 right. We got to see that answer sheet afterwards, so I checked it out. Some of those problems were easy I just approached it wrong. Others were probably just as easy, accept I never grasped the concept. Did you take the AMC 12? On 16 I realized after the test I could have measured their drawing to get the answer

Oh and on 22, what does inscribed mean?

Heres that question and another if anyone else can explain

22. A rectangle P is inscribed(?) in a sphere of radius r. The surface area of P is 384, and the sum of the lengths of its 12 edges is 112. What is r?

a. 8 b. 10 c. 12 d. 14

...Would R be half of the length of the rectangle? I didnt go on because I wasnt sure what exactly "inscribed" entailed

Inscribed means all of the vertices of the rectangle lie on the sphere. (interestingly, there are two interpretations of this, depending on how technical you get). The grade-school interpretation should be fine, though. How does a rectangle get 12 edges ? Do they mean a rectangular parallelepiped ?

25. Let S be the set of all points with coordinates (x,y,z), where x,y, and z are each chosen from the set {0,1,2}. How many equilateral triangles have all of their vertices in S?

a. 72 b. 76 c. 80 e. 88

The points generated by the set create a discrete cube of 3^3 points. There is only a certain type of triangle formed in this discrete cube that is equilateral (Take a look at the little 1x1 cubes making up the 2x2 cube).

 

[t!m]

Haha Hessam, same here. I'm especially angry because one of my two wrong was the one about windows. I read the question as "how much money do they save with the 'buy four get one free' deal in comparison to without the deal." Such a simple one, too, but I read it too fast. And what's this 12B test?

 

[cdhotfire]

That happened to me also last year, i hate when that happens, just don't worry, ull hopefully do better next year.

 

[t!m]

Nah, senior year.

 

[cdhotfire]

hehe, junior here. . Well, u can take Contest B, i think, later on, check with ur school, today they gave Contest A. I just can't get over how hard it is, . Maybe if i post some of the questions here pep will awnser them.

 

[Hessam]

I think a good thing to know for this test is dice, they love putting problems about dice, this year like 3, last year they put some also. The hardest one i did was #15, anyone get the awnser to that one, i got E. Also i hate when they ask u how many of something there are, I am like what the hell u want me to take an hour counting them, grrr. Like #11, why in the world would i spend the whole test time on that question, onless anyone knows and easy way out.

Last year my friend got 18/18, but this kid was a monster, he was so crazy while he was taking it he drank a whole Mountain Dew bottle, a 2 liter one!

15 was (C) 1/3, I'm pretty certain of it too... i'll edit this post later w/ a proof (dont have time right now)

oh yes and for 11 there's somewhat of a simple arithmetic sequence to it

you know these are correct answers

123...234... etc
135...246... etc
147...258.. etc. and so forh and so forth

they follow the order 7 5 3 1 = 16, and count for their reverses = 32, then add all the triple numbers (111, 222,333) 32+9 = 41... which i believe was the correct answer

NOTE i did not answer this question, so feel free to correct me

 

[Hessam]

How does a rectangle get 12 edges ?

its referring to its side lengths... like the corners... consider this rectangle, it has 4 sides on a plane, then another 4 on a plane, then the 4 "lines" or sides that connect them to create the actual rectangle...

and if that doenst help, by sides it means lines between two points of vertices of the rectangle...

 

[t!m]

11 (3 digit mean) was E. 45. I think your way misses answers ending in zero: 210, 420, 630, and 840 should account for the difference of four.

 

[cdhotfire]

I can't believe this, on #15 i though it said the radius was 2AC. Nooooooooo! . Grrr, darn it. Thats one wrong i got for certain. Hessam how did u see the pattern in #11, gj.

 

[tongos]

the radius one goes like this...
with x,y,z being the sides

2xy+2xz+2yz=384
4x+4y+4z=112
xy+xz+yz=192
x+y+z=28

(x+y+z)^2=28^2

(x+y)^2+2z(x+y)+z^2=28^2

x^2+y^2+2xy+2zx+2zy+z^2=28^2

x^2+y^2+z^2+2(192)=784
x^2+y^2+z^2=400

we are trying to find (1/2x)^2+(1/2y)^2+(1/2z^2)

so it is equal to 100 and the sqrt of this is equal to 10. radius equals ten.